310 
PROFESSOR SYLVESTER AND MR. J. HAMMOND 
Resuming the development of (p — q) in terms of (q — r), we have 
Order 1. 
r 
1 
3 
r~ 
(7 - ! f 
(?->')* 
~~v~ 
II 
1 
— 2(9/ + 2 q*r — q + - 
O 
+ 3 0\T* — S 3 
3 
4- 
_ 1 n -i T 2 _j_ _5_ T I U, Q 2 1 _ T 
2 q ' ni 8 r n^i 2 6 ^ 1 
<T 3. 
" 4- 
The terms of order inferior to § are of no value for present purposes, and are only 
retained for the benefit of those who may wish to carry on the work. 
To reduce the terms of Order 1, we write, in succession, 
q — (r — 0 i \/r) 2 , 
0 lX /r — + f 6. z \/s — e, 
r=(s— 0 2 \/ s) 2 . 
Thus 
S ~ — 20 2 r 2 -f- 2 cpr — q 
= J - 2 9 2 r 2 + 2r 2 - r 2 ; - 20 / + ^ 0 / ; - ^ 0 2 r 
O 
= I - f - 20, V ; + v ; - V * A- 
= f-5-2r(| + f; + 4o-(| +1<v;) + ¥V; -20— Ye,v 
= S - 4 - 1 / - 40/ + 60/ 3 - 40/ + 0/ 3 ) - | S 2 / + 4(9./ + 40/) 
O 
+ I 0/ (.s 3 4- 4c?/ + 40/) — f dps- / 4- 419/ 4- 40/); 
+ | ers 4 - hr 0/ ; 4- f e0 8 r \/s — 2 ^’ — AT 6\ r 
= I ers 4 - / <V 4 
4-1 0 a W 4- | - 0/ 3 - 2eV - V- ^ 
Hence 
Order f 
<* 3 
5 5 \ Aa • 
11 
(p - <l) " 
r 
(q - rf 
(q - rf 
ii 
1 
10 / ! — S 3 + I er.s 4 - / 0 / 
Order § 
T + 10 a ¥ + f e0/ 
lp-V 3 4- &r+ 4i-s 
s 
4 
ii 08 _ T _ 0* s t _ 2 e 3 7 ’ - / 0 / „ < f. 
